GPUs are ubiquitous in modern computers. Following are NVIDIA GPUs on today’s typical computer systems.
NVIDIA GPUs
H100 PCIe
RTX 6000
RTX 5000
Computers
servers, cluster
desktop
laptop
Main usage
scientific computing
daily work, gaming
daily work
Memory
80 GB
48 GB
16 GB
Memory bandwidth
2 TB/sec
960 GB/sec
576 GB/sec
Number of cores
???
???
???
Processor clock
??? GHz
??? GHz
??? GHz
Peak DP performance
26 TFLOPS
??? TFLOPS
??? TFLOPS
Peak SP performance
51 TFLOPS
91.1 TFLOPS
42.6 TFLOPS
2 GPU architecture vs CPU architecture
GPUs contain 1000s of processing cores on a single card; several cards can fit in a desktop PC
Each core carries out the same operations in parallel on different input data – single program, multiple data (SPMD) paradigm
Extremely high arithmetic intensity if one can transfer the data onto and results off of the processors quickly
3 GPGPU in Julia
GPU support by Julia is under active development. Check JuliaGPU for currently available packages.
There are multiple paradigms to program GPU in Julia, depending on the specific hardware.
CUDA is an ecosystem exclusively for Nvidia GPUs. There are extensive CUDA libraries for scientific computing: CuBLAS, CuRAND, CuSparse, CuSolve, CuDNN, …
The CUDA.jl package allows defining arrays on Nvidia GPUs and overloads many common operations.
The AMDGPU.jl package allows defining arrays on AMD GPUs and overloads many common operations.
The Metal.jl package allows defining arrays on Apple Silicon GPU and overloads many common operations.
AppleAccelerate.jl wraps the macOS Accelerate framework, which provides high-performance libraries for linear algebra, signal processing, and image processing on Apple Silicon CPU. This is analog of MKL for Intel CPU.
The oneAPI.jl package allows defining arrays on Intel GPUs and overloads many common operations.
I’ll illustrate using Metal.jl on my MacBook Pro running MacOS Sequoia 15.4. It has Apple M2 chip with 38 GPU cores.
versioninfo()
Julia Version 1.11.5
Commit 760b2e5b739 (2025-04-14 06:53 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 24 × 13th Gen Intel(R) Core(TM) i7-13700
WORD_SIZE: 64
LLVM: libLLVM-16.0.6 (ORCJIT, alderlake)
Threads: 1 default, 0 interactive, 1 GC (on 24 virtual cores)
usingBenchmarkTools, LinearAlgebra, RandomRandom.seed!(257)n =2^14# on CPUx =rand(Float32, n, n)y =rand(Float32, n, n)z =zeros(Float32, n, n)# on GPUxd =oneArray(x)yd =oneArray(y)zd =oneArray(z);
6.1 Dot product
# SP matrix dot product on CPU: tr(X'Y)bm_cpu =@benchmarkdot($x, $y)
BenchmarkTools.Trial: 88 samples with 1 evaluation per sample.
Range (min … max): 54.340 ms … 59.727 ms┊ GC (min … max): 0.00% … 0.00%
Time (median): 57.093 ms ┊ GC (median): 0.00%
Time (mean ± σ): 57.037 ms ± 1.232 ms┊ GC (mean ± σ): 0.00% ± 0.00%
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54.3 ms Histogram: frequency by time 59.6 ms <
Memory estimate: 0 bytes, allocs estimate: 0.
# SP matrix dot product on GPU: tr(X'Y)# why are there allocations?bm_gpu =@benchmark oneAPI.@syncdot($xd, $yd)
BenchmarkTools.Trial: 10 samples with 1 evaluation per sample.
Range (min … max): 529.678 ms … 533.094 ms┊ GC (min … max): 0.00% … 0.00%
Time (median): 531.661 ms ┊ GC (median): 0.00%
Time (mean ± σ): 531.526 ms ± 1.059 ms┊ GC (mean ± σ): 0.00% ± 0.00%
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530 ms Histogram: frequency by time 533 ms <
Memory estimate: 153.11 KiB, allocs estimate: 1877.
# speedup on GPU over CPUmedian(bm_cpu.times) /median(bm_gpu.times)
0.10738569883334086
6.2 Broadcast
# SP broadcast on CPU: z .= x .* ybm_cpu =@benchmark$z .=$x .*$y
BenchmarkTools.Trial: 54 samples with 1 evaluation per sample.
Range (min … max): 89.552 ms … 100.786 ms┊ GC (min … max): 0.00% … 0.00%
Time (median): 93.440 ms ┊ GC (median): 0.00%
Time (mean ± σ): 93.341 ms ± 1.816 ms┊ GC (mean ± σ): 0.00% ± 0.00%
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89.6 ms Histogram: frequency by time 95.8 ms <
Memory estimate: 0 bytes, allocs estimate: 0.
# SP broadcast on GPU: z .= x .* y# why is there allocation?bm_gpu =@benchmark oneAPI.@sync$zd .=$xd .*$yd
BenchmarkTools.Trial: 43 samples with 1 evaluation per sample.
Range (min … max): 116.423 ms … 119.207 ms┊ GC (min … max): 0.00% … 0.00%
Time (median): 117.007 ms ┊ GC (median): 0.00%
Time (mean ± σ): 117.182 ms ± 571.900 μs┊ GC (mean ± σ): 0.00% ± 0.00%
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116 ms Histogram: frequency by time 119 ms <
Memory estimate: 30.91 KiB, allocs estimate: 289.
# SP matrix multiplication on GPUbm_gpu =@benchmark oneAPI.@syncmul!($zd, $xd, $yd)
BenchmarkTools.Trial: 1 sample with 1 evaluation per sample.
Single result which took 10.388 s (0.00% GC) to evaluate,
with a memory estimate of 320 bytes, over 13 allocations.
For this problem size on this machine, we see GPU achieves a staggering 9 TFLOPS throughput with single precision!
# SP throughput on GPU(2n^3) / (minimum(bm_gpu.times) /1e9)
8.467424104597974e11
# SP matrix multiplication on CPUbm_cpu =@benchmarkmul!($z, $x, $y)
BenchmarkTools.Trial: 1 sample with 1 evaluation per sample.
Single result which took 12.125 s (0.00% GC) to evaluate,
with a memory estimate of 0 bytes, over 0 allocations.
# SP throughput on CPU(2n^3) / (minimum(bm_cpu.times) /1e9)
7.254477997545334e11
We see >10x speedup by GPUs in this matrix multiplication example.
# cholesky on Gram matrix# This one doesn't seem to work on Apple M2 chip yetxtxd = xd'xd + I@benchmark oneAPI.@synccholesky($(xtxd))
BenchmarkTools.Trial: 1 sample with 1 evaluation per sample.
Single result which took 7.649 s (0.00% GC) to evaluate,
with a memory estimate of 1.00 GiB, over 3 allocations.
We don’t see GPU speedup of Cholesky at the moment.
7 Evaluation of elementary and special functions on GPU
7.1 Sine and log functions
# elementwise function on GPU arraysfill!(yd, 1)bm_gpu =@benchmark oneAPI.@sync$zd .=log.($yd .+sin.($xd))bm_gpu
BenchmarkTools.Trial: 32 samples with 1 evaluation per sample.
Range (min … max): 156.100 ms … 159.967 ms┊ GC (min … max): 0.00% … 0.00%
Time (median): 157.062 ms ┊ GC (median): 0.00%
Time (mean ± σ): 157.240 ms ± 910.113 μs┊ GC (mean ± σ): 0.00% ± 0.00%
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156 ms Histogram: frequency by time 160 ms <
Memory estimate: 30.91 KiB, allocs estimate: 289.
# elementwise function on CPU arraysx, y, z =collect(xd), collect(yd), collect(zd)bm_cpu =@benchmark$z .=log.($y .+sin.($x))bm_cpu
BenchmarkTools.Trial: 3 samples with 1 evaluation per sample.
Range (min … max): 2.368 s … 2.381 s┊ GC (min … max): 0.00% … 0.00%
Time (median): 2.378 s ┊ GC (median): 0.00%
Time (mean ± σ): 2.376 s ± 6.592 ms┊ GC (mean ± σ): 0.00% ± 0.00%
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2.37 s Histogram: frequency by time 2.38 s <
Memory estimate: 0 bytes, allocs estimate: 0.